Research

The normal covering number of a finite group is the smallest number of proper subgroups whose conjugates cover the group. By a theorem of Jordan, this number must be at least two for any finite group.

Building on the work of Britnell and Maróti on the general linear group, I am investigating bounds on the normal covering number of the symplectic, orthogonal and unitary groups.

High quality datasets such as the Demographic and Health Surveys (DHS) can be used to model child poverty and inform organisations and policymakers where the need is most. However these surveys are expensive and time consuming to collect, and can quickly become outdated, making any models less effective into the future. Improvements to computer-vision models alongside easy access to Landsat and Sentinel satellite data offer a way of extending these models and potentially improving temporal and spatial estimates of child poverty.

In this paper we assess the effectiveness of computer-vision models including DINOv2, and satellite specific models such as MOSAIKS and SatMAE in predicting the level of child poverty across nineteen countries in Africa.

Choose two elements at random from a finite group, what is the probability that they commute? This probability, called the degree of commutativity, measures the abelianess of a group. It turns out, if a group is non-abelian this probability is at most 5/8 and this value is achieved by the dihedral and quaternion group of order eight.

In this expository report, I explore how the degree of commutativity is linked to properties such as solubility and nilpotence. I then look to isoclinisms, a relation between groups under which the degree of commutativity is preserved. From this I show that there are a finite amount of values the degree of commutativity can achieve strictly between 11/32 and 1/2. Furthermore I show there is a countable family of values the degree of commutativity can achieve above 1/2. For some of these values, I show that they uniquely determine the isoclinism class of that group.

Finally, I present an analogue of the degree of commutativity for infinite groups originally introduced by Antolín, Martino and Ventura, and prove some analogous results to the finite case.

Typically, to prove a number α is irrational, you construct a sequence that converges suitably quickly to that number. However in the simply stated Brun's Irrationality Criterion, you instead show there is a sequence converging to α that satisifies some interesting geometric properties, with no mention of α. This criterion, although interesting, is difficult to use in practise.

Nonetheless, in 2014, Butler used it to show ζ(3) was irrational. Based on Butler's work and using sequences by Apery, I use this geometric criterion to show that ζ(2) is irrational.